15b_00f_qs - 15B Weekly Quizzes Fall 2000 Q1 A family has 4...

15B Weekly QuizzesFall 2000Q1. A family has 4 girls and 3 boys.(a) How many ways can they sit in a row?Ans:7!.(b) How many ways can they sit in a row if boys and girls must alternate?Ans:4!×3! since the only possible seating pattern in GBGBGBG so we seatthe boys (4!) and seat the girls (3!).Q2. How many ways cantteams each of sizesbe made fromstpeople? The teams haveno names or other distinguishing features. Three versions were given depending onstudent ID number:s=2,t=4;s=3=3;s=4.Ans:One could label the teams (count ordered teams). This will bet! times the num-ber of teams since there aret! ways to assigntlabels totdiﬀerent sets of people. Thenumber of labeled teams is(sts,s,...,s)and so the answer is(st)!t!(s!)t. (One can do this withrepeated binomial coeﬃcients instead of multinomial coeﬃcients:(sts)(s(t-1)s)···.)Q3. LetA,BandC⊆Bbe sets. We makeBAinto a probability space by selectingfunctions fromAtoBuniformly at random.(a) What is the probability that a randomfis an injection?Ans:Since there are altogherbafunctions, each has probability 1/ba. An injec-tion is ana-list without repetition fromB, so there are (b)aof them. Thus theprobability is (b)a/ba.

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